A note on ill-posedness for the KdV equation

TL;DR

This paper demonstrates that the solution map for the KdV equation is not continuously extendable in Sobolev spaces with regularity below -1, using the Kato smoothing effect and Miura transform.

Contribution

It establishes a new ill-posedness result for the KdV equation in low regularity Sobolev spaces, highlighting limitations of solution continuity.

Findings

Solution map cannot be extended in H^s for s < -1

Uses Kato smoothing effect and Miura transform

Identifies thresholds for well-posedness

Abstract

We prove that the solution-map $ u_0 \mapsto u $ associated with the KdV equation cannot be continuously extended in $ H^s(\R) $ for $ s<-1 $. The main ingredients are the well-known Kato smoothing effect for the mKdV equation as well as the Miura transform.